From the nested interval theorem, the intersection of nested closed intervals is a singleton set, or another closed interval. Continue selecting nested closed intervals until the intersection is a singleton set. This smallest intersection interval consists of three points, the rational closing points and an interior singleton point. The rationals are not continuous. The singleton point cannot be rational.

Thanks, but by assuming Rationals are not continous aren't we somewhere using what we have to prove. I feel so.
Thanks

I am not sure what proofs and definitions you have at you disposal to work on this.

Here is a discussion of the Dirichlet function that shows that it is possible to find an irrational number as close as desired to a rational number. Possibly the article's provided δ ε approach will show you how to prove what you need.

You don't need to use "rationals aren't continuous". (In fact, only functions are continuous. What is meant is that "no non-singleton set in the set of rationals is connected".)

As aleph1 said, the intersection of any such collection of intervals consists of a single point. It should be obvious from that is in everyone of those intervals and so in the intersection. Since there is only one point in the intersection, that point is which is not rational.